Portfolio Selection under Condition of Variable Weights

Portfolio Selection under Condition of Variable Weights

Reza Keykhaei, Mohammad Taghi Jahandideh

College of Mathematical Sciences, Isfahan University of Technology, Isfahan, Iran Email: [email protected], [email protected]

Received July 11, 2012; revised August 9, 2012; accepted August 16, 2012

ABSTRACT

In this paper we generalize the single-period Markowitz Mean-Variance portfolio selection problem. The Markowitz’s model requires that after choosing the number of each security which constructs the portfolio in the beginning of the investment period, these numbers remain constant during and at the end of the investment period. We drop this assump- tion and consider an investment model in which the number of each security can vary randomly during the investment period. Indeed we consider a single-period investment with the property that the initial weight of each security is not equal to the final weight of that security. We redefine the notion of the rate of return of each security and show that the return of the investment in a cash account is not certain. We investigate some alternatives among risky securities which acts similar to cash accounts. For this we introduce the notion of free security and relate free securities to a riskless se- curity.

Keywords: Mean-Variance Optimization; Riskless Asset; Efficient Frontier; Free Asset

1. Introduction

After that Markowitz [1] originated the Modern Portfolio Theory (MPT) by introducing his single-period Mean- Variance (M-V) portfolio selection problem in 1952, many authors have worked on this field. Markowitz considered a portfolio of risky asset. In 1958 Tobin [2] introduced the notion of riskless asset when he entered cash account in his portfolio and stated the one-fund theorem. This theorem says that when both borrowing and lending at the riskless rate are allowed, there is a single risky fund such that every efficient portfolio (portfolios whose expected returns are largest among all portfolios with the same variance) can be constructed as a combination of the fund and the riskless asset. In the Mean-Standard Deviation space, the efficient portfolios can be represented by a straight line which is called the optimal Capital Allocation Line (CAL) (see Tütüncü [3]). In 1972, Merton [4] described the mathematics of the efficient portfolios in full analytic approach. The geometric viewpoint of the Mean- Variance analysis was developed by Merton [4] and Roll [5]. There are a huge contributions about asset allocation problem. Recently Steinbach [6] in a survey paper has a complete review on the different models in this case.

In the classic one-period M-V portfolio selection prob- lem, the weight assigned to each security contained in portfolio has been considered to be constant. In other words, if someone at the beginning of the investment period (t = 0) enters in the capital market with initial

wealth X

 

0 and invest his/her wealth as

 

0 = =1

 

0

n i i i

X



S

between n securities, then his/her wealth at the end of the investment period (t T= ) is

 

= ni₌₁i i

 

X T



S T ,

where Si

 

0 and S Ti

 

are, respectively, the value of

the ith security at the beginning and the end of the period, and for each i, i is the number of shares of the ith security. Here, S Ti

 

is nondeterministic and is con-

sidered to be a random variable, although, at the beginning and at the end of the investment period, i’s remain fixed.

In our model, we assume that i’s are nonconstant and vary randomly. In fact it is possible that i

 

0 i

 

T

for some i, where for each i, is the number of shares of the ith security at the beginning of the period and

 

0

i



 

i T

 is the number of the same one at the end of the period. Thus the value of portfolio at the beginning of the period is

 

0 = =1

   

0 0

n i i i

X



 S

and at the end of the period, it is

=1

( ) = in i( ) ( )i X T



 T S T .

reward, during the investment period (after constituting his initial portfolio), it is possible that he/she loses some amount of the ith security by a contract that force him/her to pay his obligation via the ith security. This variability is more plausible in the commodity markets. One who intends to invest his/her wealth on the com- modities, some activities such as holding and trans- porting of commodities impose additional risks; besides other commodity risks, caused by natural or accidental events, on the investment. If we consider the occurrence of the above events as a possibility, then the value of

will be a random variable.

 

i T



The importance of our model can be seen in the following conditions:

 There is someone who is willing or is forcing to make an investment by imposing some obligations in his/her portfolio investment;

 There might be no possibility or willing to insure the commodities included in the investment;

 Moreover, since this model generalizes the Markowitz model, we can use the obtained results in the classic portfolio selection problems.

It is important to note that although the essence of , as a random variable, is common between all investors, but the nature of i

 

S T

 

i T

 is special for each investor and can vary from one investor to another. Thus we fit our model to a special investor who wants to construct the optimal portfolio by considering the fact that the security weights can vary randomly.

Under this new assumption, we want to develop the notion of one-period Markowitz portfolio selection problem. On the other hand, we know that in the classic Mean- Variance analysis (with invariable weights), the rate of return of each security i and the rate of return of the portfolio r are defined. Then the mean and the variance of r are set the criteria for choosing the optimal portfolio. Indeed and r are defined as follow:

r

i r

 

 

 

0

= , =

0

i i

i

i

S T S

r i

S 



1, ,n (1)

and

 

 

 

0

=

0 X T X r

X 

, (2)

then

 

 

 

 

 

 

 

 

=1 =1

=1

=1

0 =

0

0 0

=

0 0

= ,

n n

i i i i

i i

n

i i i i

i i

n i i i

S T S

r

X

S S T S

X S

x r

 

 











in which

 

 

0

= , = 1,

0

i i i

S

, .

x i

X 

 n (3)

Clearly

=1 = 1

n i i x



.

Indeed for each i, xi is the the assigned weight allocated to the ith security in the portfolio. Here we denote each security by its weight and the portfolio by the vector of security weights. Thus xi and



1



: = , , ' n n x x R

X 

denote the ith security and the portfolio, respectively. Obviously in the case that i’s are invariable, xi’s are also invariable and for variable i’s, xi’s are variable. The M-V portfolio selection problem minimizes the variance of r, subject to a desired mean return ρ. Indeed the problem leads to the following quadratic programming problem which its solution is known:

Problem

 

 .

1 _' min

2 X

X X

. . ' = 1, ' = , s t



1 X

R X

where



is the covariance matrix of the security re- turns and R is the vector of mean returns



r1, , rn



'.

Also 1 is the n-column vector of ones.

This paper is organized as follows: In Section 2, we introduce the model and its formulation. The connection between free securities and riskless assets is investigated in Section 3. In Section 4, we describe the relation be- tween a set of free securities with both a free security and a riskless security.

2. Portfolio with Variable Weights

Now we want to know how we should state the portfolio selection problem with variable weights so that we can apply the standard methods to solve it (see Theorem 2.5). To do this, we show that, if an investment asset has any additional benefits beyond capital gains such as random growths or losses of shares, then returns should be adjusted to include any additional components of returns. As the first step, it is necessary to determine the in- crements of the number of shares for each security.

Definition 2.1. For a portfolio with variable weights,

let for each i,

 

 

 

0

= ,

0

i i

i

i

T

 



 

at the beginning and at the end of the period, respectively. We call i the rate of increments of xi.

Example 2.2. Assume that an event occurs according

to a Poisson process with rate λ, and for any arrival, the investor obligate to pay one share of the

ith security, where i is fixed. Then at the end of the investment period , with ,

 



N t t: 0



t T=





 

= ,1, 2, k

N T k

 

=

 

0 , = 0 .

i T i k

   

Thus i is a nonpositive random variable such that







 



 





 

 

= = = 0 

= =

= ,

!

i i i

k T

P l P T k

P N T k

T e

k



 





for = i

 

0

By applying the above definition and (2), we have l k  , where k= 0,1, 2,.



   

 



 

   

 

 

 

 

   

 

 

 





 

 













=1 =1

=1

=1

=1 =1

=1

=

0

0 0

=

0 0

0 0 0

0 0 0

= 1

= 1

n n

i i i

i i

n

i i i

i i

n

i i i i

i i i

n n

i i i i i

i i

n

i i i i

i

T S T r

X S S T

X S

S T S T

X S

x r x r

x r r

 



  





 



 

 

 















0 0

0

,

i

i

i

S

S

for

   





 

= 0 0 0

i i i

x  S X

as the initial weight allocated to the ith security. r can be written as

=1

= ni i i r



xR, where





:= 1

i i i i

R r r .

Definition 2.3. Let for each , i be the rate of the return of the security

= 1, ,

i 

i

n r

x . We define ui:= 1Ri and

=1

:= in i i R



xu

as the total-return of xi and the total-return (or briefly the return) of the portfolio, respectively.

Note. Suppose we possess one share of the ith security.

By considering its total-return , at the end of the period, its value becomes i

u

 

0 =

 

0 1



1

i i i i i

S u S  r



,

as expected, since 1i is the number of this security. Also, X T

 

=RX

 

0 .

Now we can state our portfolio selection problem in the form of the problem

 

 . Assume that all the se- curities are risky and is the n-column random vector and

U

*

 is the covariance matrix of the totalreturns

1, , n. Also, define

u u U:=

 

U as the n-column

vector of mean total-returns u1, , un of these n securities. The mean and covariance matrix of the total- returns are assumed to be known. By these assumptions, the portfolio selection problem with variable weights can be stated in the following form:

Problem 1. Consider the desired mean total-return ρ

for the portfolio. Our aim is to solve the following Mean-Variance portfolio selection problem,

*

1 _' min

2 X

XX

. . ' = 1, ' = . s t



1 X

U X

Assumptions. The imposed assumptions on the

problem are as follows:

 (A.1) The covariance matrix



* is positive definite.  (A.2) The mean vector U is not a multiple of 1.

The following constants are frequently used in the sequel and we define them similar to those defined in [6] as follows:

* 1 * 1

* 1 2

:= ' , := ' ,

:= ' , := .

 

   

 

 _

1 1 1 U

U U

 

 (4)

Lemma 2.4. The constants α, γand δare positive.

Proof. See Lemma 1.3 of [6].

In the above lemma we realized that we can not specify the exact rang of β. Indeed βmay not be positive.

By applying the Lagrangian multiplier method, we can interpret the solution of Problem 1. In the following we display the primal-dual solution of Problem 1 by



*_{, , }* *



X . Also  2

 

denotes the optimal variance

of the problem, which is the variance of the solution. Like [6], we call the whole graph of the optimal variance the efficient frontier.

Theorem 2.5. Problem 1 has the unique primal-dual

solution













* * 1 * *

*

*

= ,

= ,

= .

 

   

   

 _





X  1 U

(5)

Also the optimal variance is

 





2 ₌ * * ₌ 2 _{2 ,}

Definition 3.2. Let the total-return of a risky security

be uncorrelated with the total-return of all securities in a set of risky securities S. We call such a security free

(with respect to S) or S-free; and display it and its

total-return by

Proof. See Theorem 1.5 and Theorem 1.7 of [6].

Note. Although it seems that the variability of the

weights complicates the portfolio selection problem, we see that the problem can be converted to the classical form of portfolio selection problem with invariable weights (Problem

 

 ) by using the notion of total- returns. Therefore, we can employ most of the results for portfolio selection in the situation of invariable weights to the case of variable weights. Clearly when

0 iid i   1

for each , then we have a portfolio selection problem with invariable weights. Thus our model can be interpreted as an extension of classical M-V optimization problem.

= 1, ,

i  n

3. Existence of Riskless Securities

Note that a security is riskless, if its return is guaranteed and hence it is deterministic. Although cash accounts are commonly considered as riskless securities, in our model they are not riskless. Because of the change in weights, the amount of the gain for a cash account is not de- terministic. Indeed the variance of its total-return is not zero. Now we redefine the notion of riskless security on the bases of its total-return.

Definition 3.1. We say a security is riskless, if the

variance of its total-return is zero and denote it by its weight

0

f

x .

By the above definition, a cash account is riskless if and only if the rate of its increments is constant. In addition, it is not necessary for a riskless security to be of the form of a cash account.

As mentioned above, in our model, it is rare that a portfolio contains a riskless security. In the following, we try to find some risky securities which act like riskless securities to improve our model. Clearly this can also be applied for M-V portfolio selection with invariable weights.

f

x and uf, respectively.

Example 3.3. We investigate the correlation coeffi-

cients between the weekly rate of returns of a set of stocks chosen from S & P500: AEP, ARG, BMY, CPB, DPS, DUK, ERTS, LH, LLY, MCD, MKC, PGN, PPL, SCG, SO, SWY, UNH, WAG, WAT, WDC, WEC, WFV, WLP, WPI, WPO, WYNN, YHOO, YUM, ZION, ZMH. The data consist of daily closing stock prices over 2 years period from 3/12/2009 to 3/11/2011. We choose two assets ARG and WEC, which have low correlation coefficients with the other assets and present the correlation coefficients between their rate of returns and the rate of returns of the other assets in Table 1.

Suppose we can construct a portfolio for which the rate of increments of each asset has a mean in



1, 0



, is independent of the rate of return of that asset and it is independent of the rate of increments and the rate of returns of the other assets. Let us say two random variables are uncorrelated if the absolute value of their estimated correlation coefficient is less than 0.1 (at a significant level grater than 0.05). Now we can observ from Table 1 that for a weekly investment, the stock

WEC is S-free, where S = {ARG, BMY, CPB, DPS, LH, MKC, PPL, UNH, WAT, WDC, WFV, WLP, WPI, WYNN, YHOO, YUM, ZION, ZMH}, since, for WEC with the total-return _w and the ith asset of S with the total return , we have

u

i

u









 







































,

= 1 1 , 1 1

= 1 1 ,

1 1 ,

= 1 1 ) , ,

w i

w w i i

w i w i

w i w i

w i w i

Cov u u

Cov r r

Cov r r r r Cov

Cov r r

 

 

 

 

   

 

  

[image:4.595.54.541.581.715.2]

 

Table 1. Estimated values of correlation coefficients.

AEP ARG BMY CPB DPS DUK ERTS LH LLY MCD

WEC 0.231** _{0.059 0.038 0.084 0.038 0.216}** _0.089* _{0.044 0.090}* _0.141** ARG 0.06 1 0.093* _0.097* _0.128** _0.099* _0.184** _{0 .042 0.086 0.073}

MKC PGN PPL SCG SO SWY UNH WAG WAT WEC

WEC −0.052 0.211** _{0.055 0.288}** _0.232** _0.104* _{0.016 0.107}* _{0.065 1}

ARG −0.019 0.024 0.183** _0.091* _{0.074 0.08 0.109}* _{0.071 0.283}** _0.059

WDC WFV WLP WPI WPO WYNN YHOO YUM ZION ZMH

WEC 0.042 0.049 0.028 0.051 0.100* 0.055 0.046 0.084 0.073 0.073 ARG 0.281** 0.165** 0.025 0.075 0.094* 0.196** 0.196 0.134** 0.132** 0.182**

where rw =

 

rw , w=

 

w , ri =

 

ri and

 

=

i i

   . So,



w, i





w, i



Cov u u  Cov r r .

Also, the stock ARG is S-free, where S = {AEP, LH, MCD, MKC, PGN, SWY, SO, WAG, WEC, WLP, WPI}. The correlation coefficients denoted by * _and ** _are

significant at a (2-tailed) level less than or equal to 0.05 and 0.01, respectively. In the other cases the significant levels are grater than 0.05.

3.1. Portfolio with Risky Securities and One Free Security

Let a portfolio consists of risky securities

1

1 n

, , n

x  x and a free security

x

_f . Ind



', f



eed the portfolio

can be presented as 1

1:= x 'Rn

X X , where

'

n



1

= x, ,x



X  . For convenience, throughout the paper,

we sperate the free securities from other risky securities and call X and xf the “risky part” and the “free part” of the portfolio and denote the portion of each part that appears in the primal solution by “risky solution” and “free solution”, respectively. Let n f be the random vector of total-returns and be their covariance matrix. Also let



u , ,u u



'

1= 1,

* 1



U

 





1:= 1 = u1, , ,u un f '

U  U 

and ₁ be the -column vectors of ones. In the sequel and

1



n1



*



U refer to the risky part of the

portfolio.

Now our situation is a special case of Problem 1 with one additional risky security x_f and it can be stated as a new problem:

Problem 2.

*

1 1 1

,

1 min

2 x f  X

X X

1 1

1 1

. . = 1, = .

s t



 

1 X U X

Assumptions. Here the assumptions are (A.1) and

 (A.3) The mean vector U1 is not a multiple of 11.

Note. We fix these assumption throughout the paper.

Let

 

2_:=

f Var uf



and

 

₂ 1

:= _f

s   .

We define the corresponding constants which are

stated in (4) by:

* 1 * 1

1 1 1 1 1 1

* 1 2

1 1 1

:= ' , := ' ,

:= ' , := .

s s

s s s

 

s s

    

 

 _

1 1 1 U

U U

 

 Obviously

* 1 * 1 1

0

= .

0 s

   

 

 

 

Now

2

= , = ,

= , =

s s

s s f

s su

su s

   

    

 

  (7) where u:=u_f and

2

:= 2

f u u

     2_,

and α, β, γ and δ are related to the risky part of the portfolio and are denoted by (4).

First, consider 2

f

 as a variable, what will happen when 2

f

 decreases to zero or equivalently when s

increases to infinity. More precisely, what is the behavior of the solution or the efficient frontier of Problem 2 when the variance of x_f tends to zero. Although in our model there is no riskless cash account, by choosing assets with small variances, we show that one can get similar results in contrast with the case that we have riskless cash account. That is, the solution and the optimal variance of the problem converge to the solution and optimal variance in the case where the portfolio contains riskless security. Then with a little connivance, instead of a riskless security, we can use a risky (free) security such that its total-return has the same mean with the cash account and its variance is close to zero. In fact if we consider 2

f

 as a variable, then the primal-dual solution and the efficient frontier are functions of the variable

2

f

 and to reach our aim, it is sufficient to show that they are continuous functions of . First, we investigate the solution of problem 2 when





.

2 _0,

f

  

2

f = 0

Theorem 3.4. Let f . Then, Problem 2 has the

primal-dual solution as follows:

2 _{= 0}







= u , = u f.

     









* 1

0

=    , xf = 1  u .

X  1 U  

Also, the optimal variance is





2

= .

f u

   

Proof. See Theorem 1.8 of [6].

Theorem 3.5. The primal-dual solution and optimal

variance of Problem 2 are continuous with respect to





2 _0,

f

   3_.

Proof. Let 2 ₀

f

  . By Theorem 2.5, the corresponding primal-dual solution of the problem for each ₌

 

2 1

f

s  

2_{This notation is similar to δ}c _{in [6], when u is considered to be a return} of a cash account.

3_{Explicitly we suppose that u is a constant.}





* 1



1 1 1

, ' =

s xf_s s s



 

X  1 U



, (8)









= , =

s s s s s s s s

          (9) Then,









* 1

= , =

s s s xf_s s s u s .

 _

X  1 U  

s

(10) Also the optimal variance is

 

2 ₌

s s .

     (11) By Lemma 2.4, s is positive. Thus the assertion holds for 2



_0,



.

f

  

2 _{= 0}



Let f . By Theorem 3.4, Problem 2 has the primal-dual solution





= u , = u f .

      (12)







* 1

0

= , = 1

X  1U x_f  u



, (13)

and the optimal variance





2

= f

u .

    (14) To verify the assertion at , substitute the values in (7) in Equations (9)-(11) and take limit when

2_{= 0}

f 

2

f  goes to zero or equivalently when goes to infinity and compar the results with Equations (12)-(14). Now the assertion holds.

s

Corollary 3.6. Let a portfolio of risky securities

contains a cash account xc for which its rate of

increments c is uncorrelated with the total-return of

the other securities. If the variance of c converges to

zero, then the primal-dual and the optimal variance of

the M-V portfolio selection problem converges to the

same values for which xc is replaced by a riskless cash

account xc with the same mean total-return.

Proof. Assume



x1, , x xn, c



is the portfolio. Also,

let rc, and uc be the rate of return and total-return of c

x , respectively. Then



















, = ,1 1

= 1 ,1 = 0,

i c i c c

c i c

cov u u cov u r r cov u

 

 

 

for i= 1, , n. Thus xc is a free asset and the assertion holds by Theorem 3.5.

From now on, suppose

0



* *



0',xf

X and  ₀2

 

are

the primal solution and optimal variance of Problem 2 when 2_{= 0}, respectively (see Equations (13) and (14)).

f 

3.2. Portfolio with Risky Securities: A Free Security and a Guaranteed Total Loss

Following section 1.3 of [6], consider a portfolio with risky securities ₁

1

n x, , xn and a security

x

l

with guaranteed total loss, i.e. . In this case the portfolio will be ul

 

ul = 0





_',



_' n1

l ˆ := x R 

l

X X , where X

is the risky part of the portfolio. Since x is not risky

then all covariances associated with xl vanishes. Then the mean and the variance of the total-return of the portfolio become U X' and , respectively. To eliminate arbitrage opportunity, we impose the additional constraint . With this conditions the problem can be stated as follows:

*

'

XX

0 l x 

Problem 3.

*

'

XX

,

1 min

2

xl X

1ˆ = '  0

1 X 1 X

U X

= ' = .

x 

0 l 

1,

l xl

. . s t

> 0

We apply the Karush-Kuhn-Tucker conditions to solve the above problem. To do this, let ηbe the multiplier of the nonnegativity constraint x .

Theorem 3.7. Problem 3 has the following unique

primal-dual solution:

1) If  , then

a) For   > , x_l = 0, => 0, and X, ,  are the same as in the primal-dual solution of Problem 1.

b) For    ,

* 1

=  , xl = 1 , = = 0.

X  U  , =

 2) If  0, then

* 1

= , = 1  , =

 , = = 0.

l

x

  _  

X  U

Note that the optimal variance in conditions 1) b) and 2) is

2

* 2 * 1

' = ' =  .





XX U U (15)

Proof. The Lagrangian associated with Problem 3 is













*

, , , 1

= ' ' 1

2

' .

l

l L x

x x  



 

l 

  

  

X

X X 1 X

U X



We can consider the system of necessary conditions for the problem as follow:

* ₀

= ,

1



 

     

0 0 0 1 0

' 1 0 0 ' 0 0 0

0, 0, = 0.

l

l l

x

x x



 

   

   

   

   

   

 _{   }

 

 

X

1 U

1

U



Solving the above system results in = and





* 1

=   _ .

X  U

Consider the condition

1

= 0 l x

= 0

 and

* 1 * 1

= , xl= 1  ' = 1 .

 _  _

U 1 U

 

X

Thus

* 1

= ' = ' = .

 _{U X} _U_U _ So

= 1 .

l

x 



 (16)

, then

If xl = 0 =



  



as in Problem 1. Then for 0,

=  .

 

   (17) hat

Note t  and δ are positive by Lemma 2.4. Now e

the proof of th Theorem:

Let  > 0, and let   > . If = 0, then xl is other h

negative by (16). On the and, = 0, then by (17) the value of λ will be negative, and hence η is positive. Also for

if xl <

   , xl = 0 yields that λ is positive and therefore  is n e, whereas when

= 0

egativ

 , xl is positive. For    , =xl = 0. So ) h ds.

Let < 0 part (1 ol

 , since we assume that ρ is nonnegative then the positivity of γ implies that ρ is greater than   . In this case for xl = 0, > 0 and th η is negative, where = 0

en as if  , then xl > 0.

Finally let  =0, then xl = 0 the ot

leads to positivity of λ

and then negativity of η. On her hand = 0 yields = 1 > 0

l

x and finally part 2) holds. Uniqu of the ollows from strong convexity of the objective function and full rank of the constraint.

In the following; we denote the

primal-eness solution f

dual solution of Problem 3 by



_Xˆ*_{, , ,  }ˆ _{ˆ ˆ}



_{; and  ˆ}2

 

_{as the optimal}

variance of the p

Now consider a portfo roblem.

lio that contains n risky assets , f

x and xl. Let









2

1 ', , ' = ˆ', '

n

l f f

ˆ = x x X x R 

denotes the portfolio. With these assumptions Problem 3

X X

has a new version as follows:

Problem 4.

*

1 1 1

, ,

1 ' min

2 x x_{l f} X

X X

2 1 1 1

1 1

ˆ

. . = ' = 1, 0

' = , l l

s t x x



  

1 X 1 X

U X

where 12 is a -column vector of ones. We

th





2 n al-dual

display e prim solution of Problem 4 by

* 1 ˆ

ˆ _{, ,ˆ ,ˆ}

s s s

  

X .

3.8

Corollary . Problem 4 has the following unique

primal-dual solution:

1) If s> u, then

a) For   > s s , xl= 0 and ˆ =s ˆs >0 and

ˆ ˆ ,xf, , s s

X

Problem 2.

are identic the prim n of b) For

al to al-dual solutio

s s    ,

* 1

ˆ ˆ ˆ

= , = 1 , = ,

ˆ ˆ

ˆ = , = = 0.

s l s s f s

s s s

s

su

x x

   



  



 _



2) If

X U

s  u, then

* 1

ˆ ˆ

= , 1 , = ˆ ,

ˆ ˆ

ˆ = , = = 0. =

s l s s f s

s s s

s

x su x

   



  





X  U

Proof. Rearrange the portfolio to



1

ˆ

X



X₁,x_l



',

where X1=



X',xf



'. Now apply Th em 3 portfoli

eor .7 for the o



X1,xl



' and use the definition of 1.

In order to apply Corollary 3.8, specification of sign of s

 is necessary. The value of β has the main role in ermining the sign of

det s. For example when  0, only part 1) of Corollary 8 gives the answer, w for negative β, both parts of the Corollary can be applied.

In the following let

3. hereas







*



0',x xl₀,ˆf₀ ', ,ˆ0 ˆ0,ˆ * *

ˆ

0

  

X

denote the primal-dual solution of Problem 4 when

2 _{= 0}

f

 which are stated as follows:

osition 3.9. If >u

Prop  , then * and

0 = 0

l

x

0 =ˆ0 > 0 and otherw

ˆ

 ise



_ˆ*_',ˆ*



0 xf₀ '

X ,

0

ˆ

 and ˆ0 w

is identical to the primal-dual solution of

Problem 2, hen 2_{= 0}

f

 . Also for u,

* *

0 ₀

*

0 0 0

0

ˆ ' = 0, = 1 ,

ˆ _ˆ

ˆ = , = ˆ = = 0.

l

f

x

u

x u



 _{  }



X

(18)

Proof. See Theorem 1.12 of [6].

Problem 4 when We show the optimal variance of

2 _{= 0}

f by

 2

 

0

ˆ

  that is

 





2

2 0

0,



ˆ =

, > .

f

if u

u if u

  

  

 



 (19)

Theorem 3.10. The primal-dual solution and optimal

variance of the Problem 4 are continuous with respect to





2 _0,

f

   .4

fine

Proof. De the function g by

 

=t u2, . g t   t R

tu  Observe that for any 2



_0,



f

   ,

 

2 ₌

f s s g    . Let < 0, then g is strictly increasing on the in rval te



 , u 



, and hence it is one to one. Also, u< s s

1). We fix ρand investigate the cla

Suppose u

(see Figure im.

 and 2



_0,



f

   . By Corollar

th u 1

y 3.8 e primal-d ution related to any ₌

 

2

f s 

al sol  is

* 1

ˆ ˆ

= , = 1 , ˆ ,

ˆ ˆ

ˆ = , = = 0

= s s l_s s s f_s s

s s s

s

x x su

   



  

 and optimal variance is

 _

X  U

 

2

2

ˆ_s = = ˆ_s. s 

  



Because of the positivity of s, ˆs and consequently each of the above terms are continuous with respect to





2 _0,

f

   . By taking limit of these terms when 2

f  goe (or equivalently when

s

goes infinity) and comparing the results with Equatio s (18) and (19) the assertion holds.

Now let >u s to zero

n

 and 2



_0,



_{. Note that}

f

  

 

1 _<

g  u  . By Corollary 3.8 the primal-dual solution is





 

 

2 1

, <

ˆ s s s if f g

2 1

ˆ = = ,

0, s s

f

if g

     

 

  

 _



 (20)



 







 

 

2 1

2 1

, <

ˆ = ,

,

s s s f

s

s f

if g

if g

     



   

 

 

 



 (21)

,

 

 

2 1

2 1

0, <

= ˆ

1 ,

f ls

s s f

if g

x

if g

 

   

 

 

 

 (22)

when  < 0.



ˆ _{ˆ ,}



2 _< 1 [image:8.595.56.291.420.716.2]

s u i g

Figure 1. The continuity of ˆ_s

 

 

2 1

= ,

ˆ ,

s s f

fs

s f

f x

su if g

   

   



 



 _

 ₍₂₃₎





 

 

* 1 2 1

* 1 2 1

ˆ _{ˆ , <}

=

ˆ ,

s s f

s

s f

if g

if g

   

  

 

 

 _

 

 

1 U

X

U



 (24)

and finally the optimal variance is

 





 

 

2 ₂ 2 1

2

2 2 1

, <

ˆ = .

,

s s s s f

s

s f

if g

if g

       

 

   







 _



(25) As we know



 



s

 is positive. thus the primal so

-dual lution and optim variance are continuous on al

 



_g1 _{ ,}



_.

Also, it is easily seen that two parts of all of the above terms coincide at _2_f =g 1

 

_{ . Now the assertion holds}

by Theorem 3.5 and in this c the proof is complete. If = 0



ase,

 , then for u, part (1).(b) of Corollary 3.8 holds. Also for >u, conditions (20)-(25) hold.

When βis positive the proof can be separated into two parts, u  or u>  .

In the first case for





u

, part 1) b) of Corollary 3.8 holds and for    , only part 1) a) of Corollary 3.8 holds. For u<  < , the primal-dual solution and optimal variance are those which have stated in Equations (20)-(25). Finally in the second case for

   and u , part 1) a) and part 1) b) of Corollary 3.8 hold, respectively. When   < <u, Equations (20)-(25) hold but with a change in two conditions of each equation. In all cases it is easy to investigate the continuity of primal-dual solution and optimal variance with respect to 2

f

 on the interval



0,



. This completes the proof.

Like Corollary 3.6 we can state the following corollary:

Corollary 3.11 Let the portfolio of Problem 3 contains

a cash account x_c for which its rate of increments _c

is uncorrelated with the total-return of the other

securities. If the variance of _c converges to zero, then

the primal-dual and the optimal variance of Problem 3

converges to the same values for which x_c is replaced

by a riskless cash account x_c with the same mean

total-return.

4. Co-Mean Total-Return Free Securities

ich In the previous section, we investigate the case in wh the portfolio contains a free security. In this section we consider a portfolio with more than one free securities

1, ,

f _fi

co-mean

and call it and its portion in the primal solution, the “free part” of the portfolio and the “free solution”, respectively.

for

4.1. Portfolio with Risky and Co-Mean Free Securities

where and are the risky and t part of the portfo , respectively. Let

free securities. Also we denote the vector weight of co-mean free securities by



1



:= , , ' i xf x_fi

X 

In the following, let



1

2

:=

j

s _{f j}



 j= 1, , i.

In this section we assume that the portfolio is





:= ', ' i i R

X X X n i ,

X

lio i

X he free

i

U and *_i are the mean

vector d the covariance matrix of total-returns of securities, respectively, and i is a



n i



-column vector of ones. So the portfolio selection problem is:

Problem 2(i).

an

1 n

*

1 min

2 i i i i



X

X X

. . = 1,

= . i i i i s t



 

1 X

U X

Note that Problems 2(1) an 2 a equivalent. The assumptions are the same as those stated in section 2.1,

re

namely (A.1) and (A.3). As before, we define the following constants:

* 1 * 1 * 1

:= , := , := ,

i i i i i i

     



1  1 1__i U_{i i} U_i_i U_i

and

i

 

2 2

:= , i := 2 .

i i i i f ir ir

        

In the following



*_{, ,}



i  i i

X and  i2

 

ariance o

d ote the primal-dual solutio mal v f roblem 2(

part of the portfolio, then

en P n and the opti

i), respectively.

Lemma 4.1. Let α, β, γ, δ and δf be the constants

related to the risky

1) i=  s1  si, 2) i =



s1  s ui



,

3)





2

1

=

i s s ui

    ,

4)  i = f f

  ,

5) i =



s1  si



f , of. The proof is obvious

ention that two portfo selection problems are

eq e optimal variance for

an

Pro .

We m lio

uivalent, if they have the sam

y desired mean return ρfor the portfolio. Equivalently this means that their efficient frontiers coincide.

Proposition 4.2. Let j i< . Problem 2(i) with the set

of co-mean free securities



xf1, , xfi



and Problem 2(j) with the set of co-mean free securities



xf₁, , x_{f j1},x_f*



(for j = 1 this set contains _*

f

x on ) are equivalent,

where

ly

 

1

* 2 i

* =

:= f = k j k

s  



s .

Precisely, in both problems dua olutions are

identical and in primal solutions, individual weights of

co

l s

mmon securities are identical and

* = .

i

= fk

f k j

x



x

Proof. Considering Lemma 4.1 and the simple fact that

1 *

i j

=1k = =1k

k s k s s





 ,

Theorem 2.5 shows that both proble ave the same dual solutions, say

ms h and i

i

 , and c sequently have th

on

e same optimal variance. To complete the proof let





*

1

= ', , '

i xf x_fi

X X  .

Then





=

f_k k i i

x s   u

for . Now



= 1, ,

k  i





*



= =

=

i i

=

f_k k i i i i k j k j

x s   u s  u







which is the optimal weight of the last free asset in Problem 2(j).

. Precisely, no free security

ap

Proposition 4.3. Problem 2(i) and Problem 2(1) are

equivalent if 2

1

pears in the primal solution of Problem 2(i), i.e.

= 0 f 



xf₂, , x_fi



= 0, ,0 ,







and both problems have identical primal- al solution.

Proof. Let . By Theorem 3.4, for

so blem 2

du

2 _{= 0}

f 

1

each free security has the following weight in the = 2, ,

i  n,

lution of Pro (i):





= ,

f_i i

x s u

= u

  

where . Now the assertion h lds.

Theorem 4.4. The risky and the dual solutions, and

mal vari ge to that of

o

Pr

converges to zero. Besides, the sum of components of the

free solution converges to

oblem 2 when 2 _{= 0}

f

 , if



_i



1





* 1

=1

= k k

s s

* 0

f

x .

where

Now the assertion follows from Proposition 4.2 and Theorem 3.5.

Proof. Let = 1j in Proposition 4.2 and construct the

dummy portfolio





'

1= ',x_f*

X X ,

 

u_f* =u

 .

Note. As indicated in Theorem 4.4, it is seen that

2

* 2 2

1

= .

1 1

f

1

f _fi



   

Obviously if more and more co-mean free securities with low variances are added to the portfolio, then 2

*

f  is

e

ean Free Securities and a Guaranteed Total Loss

ence

of c hen

more and more close to zero (this is the advantage of a set of co-mean free securities with respect to on free security). Consequently the portfolio selection problem becomes equivalent to the problem in which the set of co-mean free securities are replaced by a riskless security with that total-return in the limit sense.

4.2. Portfolio with Risky and Co-M

In this section, we investigate the effect of the pres o-mean free securities in portfolio selection w portfolio consists of n1 risky securities and a total loss security. As before let X and X_i be the risky and

the free part of the olio, respectively. Now the portfolio is





portf





1

ˆ _{:= ', , ' =} ˆ_', n i

i xl R

 



X X X_i X X_i

and the problem can be stated as:

Problem 4(i).

* ˆ

1 min₂ i i i

i



X

X X

1ˆ

. . ' = = 1, 0

= , i i i i l l

i i

s t x x



 



 

1 X 1 X

U X

where is a -column vector of es. Note that Pr i1 s 4( equivalent. We display the

1

oblem dua



n i 1



1) and 4 are on and the

on

primal- l soluti optimal variance of Problem

 

4 i by



_ˆ*_{, , ,ˆ ˆ ˆ}



i   i i i

X and  ˆi2

 

, respectively.

Theorem 4.5. Let j i< . Problem 4(i) with the set of

securi



,



1,

f _fi x  x

and Problem 4(j) with the set of free securities



xf₁, , x_{f j1},x_f*



(when j= 1, the set contains only x_f*) are equivalent,

where

 

1

*_:= 2 ₌ i

* k j= k

f

s  



s .

Precisely, in both problems dual solut are identical

and in primal solutions, individual we s of common

ions ight securities are identical and

* = .

i

=j fk f

k x



x

Proof. Let





*

1

ˆ = ', ,i x xl f ,xf_i '

X X  .

and    _{i i} , then by T orem 3.7, we

free ties

If _i> 0 he

have









* 1

1 1

ˆ ˆ

= , x_l = 1 _i ,

X U

ˆ

, = , , ,

i

f f_i i i

x x u s s







 



ˆ ˆ

ˆ =i , i= i = 0 i



  

 ,

and

 

2

2

ˆi = . i   



It can be easily seen that by Lemm nd the simple fact that

a 4.1 a

1 *

=

i j

k k

=1 =1

k s k s s









,

= j i

  and j =i

we have . N e proof is

straight f part of T rem 3.7. For mal s of Prob

ow th heo orward by 1) b)

instance in pri olution lem 4(j)

* *

*

= =

ˆ ˆ ˆ

= _j= _i= i _{k i} = i . f

k j k j fk x s u s u



s u



x

0 i   . The claim can be proved similarly for

If =i j > 0 and >  i i = j j, the assertion 3.7 plied to Problems 4(i) and d Proposition 4.2.

free security ap

follows from part 1) a) of Theorem ap 4(j) an

Theorem 4.6. Problem 4(i) and Problem 4(1) are

equivalent if 2

1 = 0

f

 . Precisely, no

pears in the primal solution of Problem 4(i), i.e.



xf₂, , x_fi



= 0, ,0 ,







solution.

Proof. The case _{i= 1} is trivial. Let _{i> 1}. Rearrange Xˆ = ', ,1



X x xl _f*



',

the portfolio of Problem 4(i), Xˆ_i, to







', 2, , fi , ,l f1



'

where

 

u_f* =u

 .

f

x x x x

X  .

u 

i

If , then by applying Proposition 3.9 for Problem 4( ), we see that the primal an ual solutions are as follows:

Now the assertion holds by Theorems 4.5 and Theorem 3.10.

d d

Note. Theorems 3.5, 3.10, 4.4 and 4.7 state that a free

security or a set of co-mean free securities which their total-returns have very low variances and the same mean with the riskless security, can be considered as a alter- native for the riskless security.



₂







= 0, , , = 0, ,0 ,

X x_f  x_fi 

1

= 1 , = ,

l f

x x

u u

 



.

In this case the optimal variance is zero and complete the proof for

ˆ _{= ˆ = ˆ = 0}

i i i

  

REFERENCES

[1] H. M. Markowitz, “Portfolio Selection,” The Journal of Finance, Vol. 7, No. 1, 1952, pp. 77-91.

this u

 .

[2] J. Tobin, “Liquidity Preference as a Behavior towards Risk, Review of Economic Studies,” Vol. 25, No. 2, 1958, pp. 65-86. doi:10.2307/2296205

If > u , then osition 3.9, ual solution of

n 3.9

Problem 4 when if

of compone ts of the free solution converges to

= 0 l x

Problem 4.3

by Prop and

ˆ ˆ =i i > 0

  and otherwise the primal-dual solution is identical to the primal-d 2(i)

[3] R. H. Tütüncü, “A Note on Calculating the Optimal Risky Portfolio,” Finance and Stochastics, Vol. 5, No. 3, 2001, pp. 413-417. doi:10.1007/PL00013542

wh prim

en 0. On the other hand, by Theorem , the solution is identical to the primal-dual solution of Problem 2(1). Now the assertion follows from Propositio .

Theorem 4.7. The risky, the total loss and the dual

solutions, and the optimal variance of Problem 4(i)

converge to that of 2

2

1=

f  al-dual

[4] R. C. Merton, “An Analytic Derivation of the Efficient Portfolio Frontier,” Journal of Financial and Quantitative Analysis, Vol. 7, No. 4, 1972, pp. 1851-1872.

doi:10.2307/2329621 = 0

f

 ,

 

_* 1





1 =1

= ik k

s 



s 

converges to zero. Besides, the sum n

[5] R. Roll, “A Critique of the Asset Pricing Theory’s Tests. Part I. On Past and Potential Testability of the Theory,” Journal of Financial Economics, Vol. 4, No. 2, 1997, pp. 129-176. doi:10.1016/0304-405X(77)90009-5

[6] M. C. Steinbach, “Markowitz Revisited: Mean-Variance Models in Financial Portfolio Analysis,” Siam Review, Vol. 43, No. 1, 2011, pp. 31-85.

doi:10.1137/S0036144500376650

* 0

f

x eo

Proof. Let j= 1 in Th rem 4.5 and construct the